Abstract
AbstractLetf:M→S1be a Morse map of a closed manifoldMinto the circle, where a Morse map is a smooth map with only nondegenerate critical points. In this paper, we classify such maps up to fold cobordism. In the course of the classification, we get several fold cobordism invariants for such Morse maps. We also consider a slightly general situation where the source manifoldMhas boundary and the mapfrestricted to the boundary has no critical points. Letg: (Rm, 0) → (R2, 0),m≥ 2, be a generic smooth map germ, where the targetR2is oriented. Using the above-mentioned fold cobordism invariants, we show that the number of cusps with a prescribed index appearing in aC∞stable perturbation ofg, counted with signs, gives a topological invariant ofg.
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